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Let W be a natural number (including zero) and [W] be the set of all natural numbers less than or equal to W.

< (2^[W] x { T, F }), ([W], T), ([W], F), ⊕, ⊗ >,
where
(xw,xz) ⊕ (yw,yz) = (xw ∪ yw, xz ∧ yz)
(xw,xz) ⊗ (yw,yz) = ( (xw ∩ yw) ∪ ( xz ? xw ) ∪ (yz ? yw) , xz ∨ yz )
z ? w = if z then w else ∅

this mess is actually a semiring. That blows my mind.

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